Dynamic phases, clustering, and chain formation for driven disk systems in the presence of quenched disorder

We numerically examine the dynamic phases and pattern formation of two-dimensional monodisperse repulsive disks driven over random quenched disorder. We show that there is a series of distinct dynamic regimes as a function of increasing drive, including a clogged or pile-up phase near depinning, a homogeneous disordered flow state, and a dynamically phase separated regime consisting of high-density crystalline regions surrounded by a low density of disordered disks. At the highest drives the disks arrange into one-dimensional moving chains. The phase separated regime has parallels with the phase separation observed in active matter systems, but arises from a distinct mechanism consisting of the combination of nonequilibrium fluctuations with density-dependent mobility. We discuss the pronounced differences between this system and previous studies of driven particles with longer-range repulsive interactions moving over random substrates, such as superconducting vortices or electron crystals, where dynamical phase separation and distinct one-dimensional moving chains are not observed. Our results should be generic to a broad class of systems in which the particle-particle interactions are short ranged, such as sterically interacting colloids or Yukawa particles with strong screening driven over random pinning arrays, superconducting vortices in the limit of small penetration depths, or quasi-two-dimensional granular matter flowing over rough landscapes.

Figure 1

(a) The average disk velocity $\langle V_x\rangle$ vs. driving force $F_D/F_p$ for a system of harmonically interacting repulsive disks in a sample with $F_p=1.0$ and $N_p=1440$ at disk densities of $\varphi =0.85$ (red circles), 0.71 (orange squares), 0.61 (yellow diamonds), 0.55 (light green up triangles), 0.43 (medium green left triangles), 0.3 (dark green down triangles), 0.25 (blue right triangles), and 0.15 (purple stars). (b) The corresponding $d\langle V_x\rangle /d{F}_D$ vs. $F_D/F_p$ curves showing a peak near $F_D/F_p=1.0$. Inset: The depinning threshold $F_c$ vs. $\varphi$, where $\varphi \approx 0.3$ corresponds to a 1:1 ratio of disks to pinning sites. (c) The corresponding cluster size $C_L$ vs. $F_D/F_p$.

Figure 2

(a) The disk positions (circles) for the system in Fig. 1 at $\varphi =0.61$ for $F_D/F_p=0.3$, showing a clustering or pile up effect. (b) The corresponding structure factor $S\left(\mathbf{k}\right)$ has a ringlike signature. (c) The driven homogeneous phase in the same system at $F_D/F_p=0.7$. (d) The corresponding $S\left(\mathbf{k}\right)$ plot from (c).

Figure 3

(a) The disk positions (circles) for the system in Fig. 1 at $\varphi =0.61$ for $F_D/F_p=1.05$, corresponding to the local maximum in $C_l$ in Fig. 1. Here the system forms a density phase separated state. (b) The corresponding $S\left(\mathbf{k}\right)$ plot contains sixfold peaks due to the triangular ordering in the dense phase. (c) The same system at $F_D/F_p=2.0$ where a moving chainlike state forms. (d) The corresponding $S\left(\mathbf{k}\right)$ shows smectic ordering.

Figure 4

(a) The disk positions (circles) for the system in Fig. 1 at $\varphi =0.3$ for $F_D/F_p=0.15$, showing the formation of small clusters. (b) The corresponding $S\left(\mathbf{k}\right)$ plot. (c) The same system at $F_D/F_p=0.6$ in the moving phase where the disk density becomes homogeneous. (d) The corresponding $S\left(\mathbf{k}\right)$ shows a diffuse or liquidlike pattern.

Figure 5

(a) The disk positions (circles) for the system in Fig. 1 at $\varphi =0.3$ for $F_D/F_p=1.05$, where the disks form chainlike patterns. (b) The corresponding $S\left(\mathbf{k}\right)$ plot. (c) The same system at $F_D/F_p=2.0$ in the moving phase where the disks form a series of chains or stripes. (d) The corresponding $S\left(\mathbf{k}\right)$ has smectic ordering. (e) Blow up of disk positions from panel (c) showing formation of 1D chains.

Figure 6

The transverse displacements $\langle \delta y^2\rangle$ obtained after $4\times {10}^6$ simulation time steps (red squares) and the diffusive exponent $\alpha$ (blue circles) vs. $F_D/F_p$ for the system in Fig. 1 at $\varphi =\left(\mathrm{a}\right)0.25$, (b) 0.3, (c) 0.43, (d) 0.55, (e) 0.61, and (f) 0.71.

Figure 7

The fraction $P_6$ of sixfold coordinated disks vs. $F_D/F_p$ for the system in Fig. 1 for $\varphi =\left(\mathrm{a}\right)0.25$, (b) 0.3, (c) 0.43, (d) 0.61, (e) 0.71, and (f) 0.85. For $\varphi =0.61$ in panel (d), the local maximum in $P_6$ near $F_D=1.0$ is correlated with the formation of the phase separated state shown in Fig. 3.

Figure 8

Schematic phase diagram as a function of $F_D/F_p$ vs. $\varphi$ for the system in Fig. 1. I: Pinned or clogged state. II: Homogeneous plastic flow. III: Density phase separated state. IV: Moving smectic or moving chain state. V: Moving polycrystalline state. VI: Moving crystal state.

Figure 9

Schematic plot of the effective shaking temperature or activity $A$ vs. $F_D/F_p$ for a sample with $\varphi =0.3$. The vertical dashed black line at $F_D/F_p=0.5$ separates the pinned phase (I) from homogeneous plastic flow (II). At $F_D/F_p=1.0$ all the particles begin to move, producing an effective shaking temperature or activity with an amplitude $A$ that decreases as $1/F_D$. There is a critical shaking activity, $A_c$ (horizontal red line), above which clustering can begin to occur, so that a phase separated state (III) appears above $F_D/F_p=1.0$. Phase III disappears above the value of $F_D/F_p$ marked by a vertical solid green line, which is determined by the point at which $A$ drops below $A_c$. At high drives where $A<A_c$, Phase IV, the moving smectic state, appears as the externally applied driving force begins to dominate the behavior of the system and the effective shaking temperature becomes unimportant.

Figure 10

(a) $\langle V_x\rangle$ vs. $F_D/F_p$ at $\varphi =0.55$ and $F_p=1.0$ for $N_p/N_d=0.0$, 0.072, 0.144, 0.216, 0.288, 0.36, 0.432, 0.504, and 0.576, from top to bottom. (b) The corresponding $d\langle V_x\rangle /d{F}_D$ vs. $F_D/F_p$ curves showing peaks near $F_D/F_p=0.5$ and $F_D/F_p=1.0$.

Figure 11

Cluster size $C_l$ vs. $F_D/F_p$ for the system in Fig. 10 at $N_p/N_d=\left(\mathrm{a}\right)0.072$, (b) 0.216, (c) 0.288, and (d) 0.432. The local peaks in (b) and (c) correspond to the formation of a density phase separated state. In panel (c) the lettering indicates the $F_D/F_p$ values represented in the real space images in Fig. 12.

Figure 12

The disk positions for the system in Figs. 10 and 11 at $N_p/N_d=0.288$ for drive values marked with letters in Fig. 11. In panels (a) and (d), red disks are part of clusters containing three or more disks, while blue disks are isolated or in a cluster containing only two disks. (a) The pinned cluster state at $F_D/F_p=0.05$. (b) At $F_D/F_p=0.3$ the moving disks form more spread out clusters. (c) At $F_D/F_p=0.6$, corresponding to the local minimum of $C_l$ in Fig. 11, a homogeneous disordered state forms. (d) At $F_D/F_p=1.05$, corresponding to the peak in $C_l$ in Fig. 11, a density phase separated state forms. At (e) $F_D/F_p=1.5$ and (f) $F_D/F_p=2.0$, the disks are in a moving chain state.

Figure 13

The disk positions for a system with $\varphi =0.55$ at $N_p/N_d=0.072$, where there is no peak in $C_l$ in Fig. 11. Red disks are part of clusters containing three or more disks, while blue disks are isolated or in a cluster containing only two disks. (a) A density phase separated state at $F_D/F_p=0.3$. (b) A moving chain state forms at higher drives, shown here at $F_D/F_p=1.5$.

Figure 14

(a) Schematic phase diagram as a function of $F_D/F_p$ vs. $N_p/N_d$ for the system in Fig. 10 at fixed $\varphi =0.55$. I: Pinned or clogged state. II: Homogeneous plastic flow. III: Density phase separated state. IV: Moving smectic or moving chain state. (b) Phase diagram for the same system at $\varphi =0.55$ and $N_p/N_d=0.288$ as a function of $F_D$ vs. $F_p$.

Figure 15

(a) The cluster size $C_l$ vs. $F_D$ for samples with $\varphi =0.55$ and $N_p/N_d=0.288$ at $F_p=0.0$ (blue circles), 0.2 (blue squares), 0.4 (green diamonds), 0.6 (orange triangles), and 1.0 (red circles). (b) $\langle V_x\rangle$ vs. $F_D-F_c$ for the velocity-force curve obtained at $\varphi =0.55$ and $N_p/N_D=0.576$. The solid line is a power law fit with an exponent of $\beta =1.6$.